Irrational Numbers and Number Sets

Existing number sets, such as rational numbers, are insufficient to represent all numbers or solve all mathematical equations, necessitating the definition of a wider number set.
To achieve a complete representation of all numbers, a complementary set to the rational numbers is introduced, which is designated as the set of irrational numbers.

A rational number is defined as any number that can be expressed as the ratio of two integers.
Any number that can be written in the form of one integer over another integer is classified as a rational number.

An irrational number is defined as any number that cannot be written as the ratio of two integers.
The set of irrational numbers is the mathematical complement of the set of rational numbers.
Rational numbers and irrational numbers are mutually exclusive sets, meaning a number cannot belong to both sets simultaneously.
Classification and Non-Example of Irrational Numbers:
- The expression $\sqrt{9}$ cannot be included in the set of irrational numbers.
- Because $\sqrt{9} = 3$ (an integer), it is inherently a rational number and therefore excluded from the irrational set.

Root expressions, specifically square roots, are fundamentally tied to the study of irrational numbers when evaluating radical terms that do not simplify into integers or standard integer ratios.